Make vertices of a mesh "match" an adjacent mesh
An approximation to this could be achieved using the Shrinkwrap modifier.
Select al the vertices on the border you want to move, press Ctrl + G, and choose Assign to New Group.
Then put a Shrinkwrap modifier on the object. For the Vertex Group choose the newly made group, and as a Target choose the object you want the border to move towards. As the mode for this case it is probably best to choose Nearest Surface Point or Nearest Vertex, though both probably do not give an optimal result.

The result varies for different shapes of a border curve. As you can see from this keyshape animation two vertices of the vertex group moved to the same vertex on the target object, leaving a slight gap next to it. (So if it works for you depends on the specific meshes you work with and the amount of manually fixing you're willing to do.)